This thesis investigates combinatorial principles from order theory and Ramsey theory with a focus on foundational aspects within the framework of reverse mathematics and computability theory. The first part focuses on order dimension theory, a classical topic at the intersection of order theory and combinatorics. Intuitively, the dimension measures how “far” a poset is from being linearly ordered. We are interested in statements that give an upper bound to the dimension of a poset in terms of the dimension of its subposets, obtained by removing one or more points. We analyze these bounding theorems, calibrating their logical strength within subsystems of second-order arithmetic. The second part examines the notion of strong indivisibility, a particular Ramsey-like property. We focus on countable structures, with particular emphasis on Cameron’s classification theorem of strongly indivisible graphs. We study this classification from the perspectives of reverse mathematics and computable combinatorics, investigating the role of induction axioms and effective constructions. In the final part, we study a very general finite Ramsey theorem, where both the sets being colored and the homogeneous set must satisfy some largeness notion. Historically, largeness notions were associated with countable ordinals and systems of fundamental sequences. To extend this approach, we develop a more flexible framework using blocks and barriers. Since the complexity of barriers can be measured by countable ordinals, we define and study Ramsey ordinals, a generalization of the well-known Ramsey numbers of classical finite Ramsey theory.

Andrea Volpi

E-mail: andreavolpi207801@gmail.com

URL: https://andreasdfghj.github.io/andreavolpi/PhD_Thesis.pdf

The abstract was taken directly from the thesis.

In this article, we deal with the classification complexity of continuous (Devaney) chaotic systems in dimensions $0,1,$ and $\infty $ using the framework of invariant descriptive set theory. We identify the complexity in dimensions $0$ and $\infty $, while in dimension $1$ we get some partial results.

More precisely, we prove the topological conjugacy relation of invertible chaotic systems on the Hilbert cube (resp. on all compact metric spaces) has the same complexity as (i.e., is Borel bireducible with) the universal orbit relation induced by a Polish group. As a consequence, this answers a recent question asked by L. Ding. We also prove that the topological conjugacy relation of invertible chaotic systems on the Cantor space has the same complexity as the universal relation induced by the group $S_\infty $. This answers a recent question by M. Foreman. Some non-trivial bounds on the classification complexity of chaotic systems on the interval and on the circle are also obtained. Namely, the lower bound is the Vitali equivalence relation, and the upper bound is the equality of countable sets of reals. This especially implies that the relation is Borel. However, the exact complexity remains unknown.

We define a natural notion of standard translation for the formulas of conditional logic which is analogous to the standard translation of modal formulas into the first-order logic. We briefly show that this translation works (modulo a lightweight first-order encoding of the conditional models) for the minimal classical conditional logic $\mathsf {CK}$ introduced by Brian Chellas in [3]; however, the main result of the article is that a classically equivalent reformulation of these notions (i.e., of standard translation plus theory of conditional models) also faithfully embeds the basic Nelsonian conditional logic $\mathsf {N4CK}$, introduced in [11] into $\mathsf {QN4}$, the paraconsistent variant of Nelson’s first-order logic of strong negation. Thus $\mathsf {N4CK}$ is the logic induced by the Nelsonian reading of the classical Chellas semantics of conditionals and can, therefore, be considered a faithful analogue of $\mathsf {CK}$ on the non-classical basis provided by the propositional fragment of $\mathsf {QN4}$. Moreover, the methods used to prove our main result can be easily adapted to the case of modal logic, which makes it possible to improve an older result [10, Proposition 7] by S. Odintsov and H. Wansing about the standard translation embedding of the Nelsonian modal logic $\mathsf {FSK}^d$ into $\mathsf {QN4}$.

In The Emperor’s New Mind [7], Roger Penrose proves a variant of the halting problem, and uses it to argue that humans have cognitive capacities beyond the computable. In this short note, I explicate his argument, and show how it fails, via a corollary of his result. My response to Penrose is in fact of a kind with a number of prior responses: he assumes human powers, that (as the corollary shows) no computer could have. However, as far as I am aware, no one has previously addressed this specific form of the argument, in the direct way that I will.

We carry out a logical analysis of a convergence proof for greedy approximation schemes in uniformly smooth Banach spaces. Though the proof is by contradiction, we are able to extract computable rates of convergence that depend on the corresponding modulus of uniform smoothness for the space. While our quantitative results represent a first proof-theoretic study of greedy approximation schemes, we use this case study more generally as an opportunity to make explicit some of the high-level proof-theoretic reasoning that enables us to transform a nonconstructive convergence proof to one where computable convergence rates are apparent, representing the proof using a series of formal derivations that are designed to capture core mathematical reasoning, as opposed to low-level proof-theoretic bureaucracy. In this way we exemplify an approach to representing the process of program extraction that might, in particular, inform efforts to formalise proof mining in proof assistants.

We introduce finite support iterations of symmetric systems, and use them to provide a strongly modernized proof of David Pincus’ classical result that the axiom of dependent choice is independent over $\operatorname {\mathrm {ZF}}$ with the ordering principle together with a failure of the axiom of choice.

This article examines the results of [6] in a purely tree-theoretic context. We show how this rephrased Chase–Freitag Lemma is sufficient to prove the Sauer–Shelah and Bhaskar Lemmas. We also use this approach to expand these two lemmas to the context of arbitrary finite label sets.

A Constructive Logic of Evidence and Truth ($\mathsf {LET_C}$) is introduced. This logic is both paraconsistent and paracomplete, providing connectives for consistency and determinedness that enable the independent recovery of explosiveness and the law of excluded middle for specific propositions. Dual connectives for inconsistency and undeterminedness are also defined in $\mathsf {LET_C}$. Evidence is explicitly formalised by integrating lambda calculus terms into $\mathsf {LET_C}$, resulting in the type system $\mathsf {LET_C^{\lambda }}$. In this system, lambda calculus terms represent procedures for constructing evidence for compound formulas based on the evidence of their constituent parts. A realisability interpretation is provided for $\mathsf {LET_C}$, establishing a strong connection between deductions in this system and recursive functions.